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import Mathlib.Data.Real.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Ring

-- HONEST-RELABEL CORRECTION (2026-09-09). The previous header claimed a "formal proof
-- of the generation index quantization theorem for the E8 -> E6 x SU(3) Coxeter / Z3
-- Calabi-Yau orbifold quotient." Neither theorem below establishes this, per
-- `Chyren_Second_Brain/50_Mathematical_Notation/derivations/D47_generation_index_audit.md`
-- (2026-08-29), whose verdict this file now reflects in place. Summary of that audit:
--
-- * `mckayArrowsPerNode : ℕ := 3` / `mckay_generation_count : mckayArrowsPerNode = 3`
--   is `3 = 3` on a hand-set literal. No McKay quiver, vertex, arrow, or bifundamental
--   representation is constructed anywhere in this file.
-- * `z3_chirality_quantization` is ALSO vacuous, and this is less obvious than the
--   above -- it is not a bare `rfl`, and it does real case-split work. D47's probe is
--   the point: the identical proof, with the physics literals 3, 3, and -3 replaced by
--   arbitrary constants 42, 0, and -17 wired into the same if-then-else shape, compiles
--   unchanged and "proves" quantization to {0, 42, -17}. The quantization is an
--   artifact of literals written into the definition, not a consequence of Z3
--   character orthogonality, a Dolbeault/Lefschetz index, or any actual orbifold
--   construction. D47 also re-ran the file's companion Python Lefschetz computation
--   (`compute_equivariant_generation_index.py`): its honest output is a non-integer
--   (-1.067), unstable to a spin-structure choice that gives +7 instead -- so even the
--   numeric claim "the index is 3" is not established by the corpus this file was
--   meant to formalize. Read D47 in full before treating either theorem as physics.
--
-- This is the same defect pattern already caught and corrected in
-- `CartanTrialityGenerations.lean` (2026-08-26), the withdrawn `RamanujanGap`/`h_gap`
-- passthrough (2026-09-04), and the withdrawn `SON_Casimir.lean` (2026-08-09):
-- substitutability failure. `assurance/HARDENING_PLAN.md` cites this file as the model
-- case for a systematic vacuity audit; it had not yet received the in-file correction
-- the other three instances already have. This file gives it that correction:
-- mechanical content retained verbatim, declarations renamed to state what they
-- actually prove.

/-!
# Z3 literal-quantization and bookkeeping — honest status

Both declarations below are arithmetic on literals written into their own
definitions. Neither formalizes `E8`, `E6 x SU(3)`, a Coxeter number, a Calabi-Yau
manifold, an orbifold quotient, a McKay quiver, a vertex, a bifundamental
representation, Z3 character orthogonality, a Lefschetz/Dolbeault index, or the
physical fermion generation count. See
`D47_generation_index_audit.md` for the substitution probe demonstrating this.
-/

namespace Chyren.IndexTheory

/--
Hand-defined predicate on integer triples: true iff `a1 + a2 + a3 ≡ 0 (mod 3)`.
Motivated by (but not derived from) the Calabi-Yau trace-zero condition on Z3
weight triples; no Calabi-Yau manifold or weight lattice is constructed here.
-/
def isCY (a1 a2 a3 : ℤ) : Prop := (a1 + a2 + a3) % 3 = 0

/--
Hand-defined integer function built from two `mod 3` case splits, with the
output literals `3`/`-3`/`0` written directly into the if-then-else. D47's probe
replaces these literals with arbitrary constants (42, -17) in the identical
proof shape and it still compiles -- the output set is chosen by the definition,
not derived from any chirality, representation, or physical field.
-/
def z3NetChirality (a1 a2 a3 : ℤ) : ℤ :=
  let gen := (if a1 % 3 = 2 ∧ a2 % 3 = 2 ∧ a3 % 3 = 2 then 3 else 0)
  let anti := (if a1 % 3 = 1 ∧ a2 % 3 = 1 ∧ a3 % 3 = 1 then 3 else 0)
  gen - anti

/--
VACUOUS (D47, substitution probe). Proves `z3NetChirality` only ever takes the
values written into its own if-then-else. This is arithmetic about the
definition given, not a consequence of Z3 character orthogonality, a
Lefschetz/Dolbeault index, or any Calabi-Yau/orbifold structure. The identical
proof closes with 42 and -17 substituted for 3 and -3, and Lean cannot tell the
difference -- that is exactly what "vacuous" means here.
-/
theorem z3NetChirality_takes_defined_literals_VACUOUS_D47 (a1 a2 a3 : ℤ) :
    z3NetChirality a1 a2 a3 = 0 ∨
    z3NetChirality a1 a2 a3 = 3 ∨
    z3NetChirality a1 a2 a3 = -3 := by
  dsimp [z3NetChirality]
  split_ifs with h1 h2 h3
  · -- h1: (2,2,2), h2: (1,1,1) -> impossible since 2 % 3 ≠ 1 % 3
    rcases h1 with ⟨h1a, h1b, h1c⟩
    rcases h2 with ⟨h2a, h2b, h2c⟩
    have : (2 : ℤ) = 1 := by
      calc (2 : ℤ) = (2 : ℤ) % 3 := by decide
      _ = a1 % 3 := h1a.symm
      _ = 1 := h2a
    revert this
    decide
  · -- h1: (2,2,2), not h2 -> net = 3
    right; left; rfl
  · -- not h1, h2: (1,1,1) -> net = -3
    right; right; rfl
  · -- neither -> net = 0
    left; rfl

/-- Hand-set bookkeeping literal. Not derived from any McKay quiver, representation
    theory, or orbifold construction. -/
def mckayArrowsPerNode_DECLARED_BOOKKEEPING_LITERAL : ℕ := 3

/-- VACUOUS (D47). Reflexivity on the hand-set literal above: proves `3 = 3`, not
    that a McKay quiver on `C^3/Z_3` has 3 incoming/outgoing bifundamental arrows
    per vertex, and not that `E8 -> E6 x SU(3)` reproduces 3 generations. No
    quiver, vertex, arrow, or representation is constructed anywhere in this
    file. -/
theorem mckayArrowsPerNode_eq_three_DEFINITIONAL_PLACEHOLDER :
    mckayArrowsPerNode_DECLARED_BOOKKEEPING_LITERAL = 3 := by
  rfl

end Chyren.IndexTheory

	import Mathlib.Data.Real.Basic
	import Mathlib.Data.Complex.Basic
	import Mathlib.Tactic.Linarith
	import Mathlib.Tactic.Ring

	-- HONEST-RELABEL CORRECTION (2026-09-09). The previous header claimed a "formal proof
	-- of the generation index quantization theorem for the E8 -> E6 x SU(3) Coxeter / Z3
	-- Calabi-Yau orbifold quotient." Neither theorem below establishes this, per
	-- `Chyren_Second_Brain/50_Mathematical_Notation/derivations/D47_generation_index_audit.md`
	-- (2026-08-29), whose verdict this file now reflects in place. Summary of that audit:
	--
	-- * `mckayArrowsPerNode : ℕ := 3` / `mckay_generation_count : mckayArrowsPerNode = 3`
	-- is `3 = 3` on a hand-set literal. No McKay quiver, vertex, arrow, or bifundamental
	-- representation is constructed anywhere in this file.
	-- * `z3_chirality_quantization` is ALSO vacuous, and this is less obvious than the
	-- above -- it is not a bare `rfl`, and it does real case-split work. D47's probe is
	-- the point: the identical proof, with the physics literals 3, 3, and -3 replaced by
	-- arbitrary constants 42, 0, and -17 wired into the same if-then-else shape, compiles
	-- unchanged and "proves" quantization to {0, 42, -17}. The quantization is an
	-- artifact of literals written into the definition, not a consequence of Z3
	-- character orthogonality, a Dolbeault/Lefschetz index, or any actual orbifold
	-- construction. D47 also re-ran the file's companion Python Lefschetz computation
	-- (`compute_equivariant_generation_index.py`): its honest output is a non-integer
	-- (-1.067), unstable to a spin-structure choice that gives +7 instead -- so even the
	-- numeric claim "the index is 3" is not established by the corpus this file was
	-- meant to formalize. Read D47 in full before treating either theorem as physics.
	--
	-- This is the same defect pattern already caught and corrected in
	-- `CartanTrialityGenerations.lean` (2026-08-26), the withdrawn `RamanujanGap`/`h_gap`
	-- passthrough (2026-09-04), and the withdrawn `SON_Casimir.lean` (2026-08-09):
	-- substitutability failure. `assurance/HARDENING_PLAN.md` cites this file as the model
	-- case for a systematic vacuity audit; it had not yet received the in-file correction
	-- the other three instances already have. This file gives it that correction:
	-- mechanical content retained verbatim, declarations renamed to state what they
	-- actually prove.

	/-!
	# Z3 literal-quantization and bookkeeping — honest status

	Both declarations below are arithmetic on literals written into their own
	definitions. Neither formalizes `E8`, `E6 x SU(3)`, a Coxeter number, a Calabi-Yau
	manifold, an orbifold quotient, a McKay quiver, a vertex, a bifundamental
	representation, Z3 character orthogonality, a Lefschetz/Dolbeault index, or the
	physical fermion generation count. See
	`D47_generation_index_audit.md` for the substitution probe demonstrating this.
	-/

	namespace Chyren.IndexTheory

	/--
	Hand-defined predicate on integer triples: true iff `a1 + a2 + a3 ≡ 0 (mod 3)`.
	Motivated by (but not derived from) the Calabi-Yau trace-zero condition on Z3
	weight triples; no Calabi-Yau manifold or weight lattice is constructed here.
	-/
	def isCY (a1 a2 a3 : ℤ) : Prop := (a1 + a2 + a3) % 3 = 0

	/--
	Hand-defined integer function built from two `mod 3` case splits, with the
	output literals `3`/`-3`/`0` written directly into the if-then-else. D47's probe
	replaces these literals with arbitrary constants (42, -17) in the identical
	proof shape and it still compiles -- the output set is chosen by the definition,
	not derived from any chirality, representation, or physical field.
	-/
	def z3NetChirality (a1 a2 a3 : ℤ) : ℤ :=
	let gen := (if a1 % 3 = 2 ∧ a2 % 3 = 2 ∧ a3 % 3 = 2 then 3 else 0)
	let anti := (if a1 % 3 = 1 ∧ a2 % 3 = 1 ∧ a3 % 3 = 1 then 3 else 0)
	gen - anti

	/--
	VACUOUS (D47, substitution probe). Proves `z3NetChirality` only ever takes the
	values written into its own if-then-else. This is arithmetic about the
	definition given, not a consequence of Z3 character orthogonality, a
	Lefschetz/Dolbeault index, or any Calabi-Yau/orbifold structure. The identical
	proof closes with 42 and -17 substituted for 3 and -3, and Lean cannot tell the
	difference -- that is exactly what "vacuous" means here.
	-/
	theorem z3NetChirality_takes_defined_literals_VACUOUS_D47 (a1 a2 a3 : ℤ) :
	z3NetChirality a1 a2 a3 = 0 ∨
	z3NetChirality a1 a2 a3 = 3 ∨
	z3NetChirality a1 a2 a3 = -3 := by
	dsimp [z3NetChirality]
	split_ifs with h1 h2 h3
	· -- h1: (2,2,2), h2: (1,1,1) -> impossible since 2 % 3 ≠ 1 % 3
	rcases h1 with ⟨h1a, h1b, h1c⟩
	rcases h2 with ⟨h2a, h2b, h2c⟩
	have : (2 : ℤ) = 1 := by
	calc (2 : ℤ) = (2 : ℤ) % 3 := by decide
	_ = a1 % 3 := h1a.symm
	_ = 1 := h2a
	revert this
	decide
	· -- h1: (2,2,2), not h2 -> net = 3
	right; left; rfl
	· -- not h1, h2: (1,1,1) -> net = -3
	right; right; rfl
	· -- neither -> net = 0
	left; rfl

	/-- Hand-set bookkeeping literal. Not derived from any McKay quiver, representation
	theory, or orbifold construction. -/
	def mckayArrowsPerNode_DECLARED_BOOKKEEPING_LITERAL : ℕ := 3

	/-- VACUOUS (D47). Reflexivity on the hand-set literal above: proves `3 = 3`, not
	that a McKay quiver on `C^3/Z_3` has 3 incoming/outgoing bifundamental arrows
	per vertex, and not that `E8 -> E6 x SU(3)` reproduces 3 generations. No
	quiver, vertex, arrow, or representation is constructed anywhere in this
	file. -/
	theorem mckayArrowsPerNode_eq_three_DEFINITIONAL_PLACEHOLDER :
	mckayArrowsPerNode_DECLARED_BOOKKEEPING_LITERAL = 3 := by
	rfl

	end Chyren.IndexTheory